A278991 a(n) is the number of simple linear diagrams with n+1 chords.

0, 1, 3, 24, 211, 2325, 30198, 452809, 7695777, 146193678, 3069668575, 70595504859, 1764755571192, 47645601726541, 1381657584006399, 42829752879449400, 1413337528735664887, 49465522112961344241, 1830184115528550306438, 71375848864779552073957

Offset: 0

Author

N. J. A. Sloane, Dec 07 2016

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..301
• E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv preprint arXiv:1601.05073 [math.CO], 2016.
• E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.

Crossrefs

Cf. A003436, A003437, A007474, A278990, A278992, A278993, A278994.

Programs

Mathematica

a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := a[n] = (2 n - 1) a[n - 1] + (4 n - 3) a[n - 2] + (2 n - 4) a[n - 3]; Table[a@ n, {n, 0, 19}] (* Michael De Vlieger, Dec 10 2016 *)

PARI

seq(N) = {
  my(a = vector(N)); a[1]=1; a[2]=3; a[3]=24;
  for (n=4, N, a[n] = (2*n-1)*a[n-1] + (4*n-3)*a[n-2] + (2*n-4)*a[n-3]);
  concat(0, a);
};
seq(20) \\ Gheorghe Coserea, Dec 10 2016

PARI

N = 20; x = 'x + O('x^N);
concat(0, Vec(serlaplace((1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x))))) \\ Gheorghe Coserea, Dec 10 2016

Formula

E.g.f.: (1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x)).

a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n+3/2). - Vaclav Kotesovec, Dec 07 2016

a(n) = (2*n-1)*a(n-1) + (4*n-3)*a(n-2) + (2*n-4)*a(n-3). - Gheorghe Coserea, Dec 10 2016

Extensions

Offset corrected by Gheorghe Coserea, Dec 10 2016